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Lecture 18Lecture 18Fourier Analysis, Low Pass y ,
Filters, Decibels
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Chapter 6F R B d PlFrequency Response, Bode Plots,
and Resonance
1. State the fundamental concepts of Fourier1. State the fundamental concepts of Fourier analysis.
2. Determine the output of a filter for a given input consisting of sinusoidal componentsinput consisting of sinusoidal components using the filter’s transfer function.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
3. Use circuit analysis to determine theytransfer functions of simple circuits.
4. Draw first-order lowpass or highpass filter circuits and sketch their transfer functions.
5. Understand decibels, logarithmic frequency scales, and Bode plots.
6. Draw the Bode plots for transfer functions of first-order filters.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Fourier Analysisy
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Fourier Analysisy
All real-world signals are sums of sinusoidal t h i i f icomponents having various frequencies,
amplitudes, and phases.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Fourier Analysisy
....)5sin(4)3sin(4)sin(4)( 000 +++= tAtAtAtvsq ωωω
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
....)5sin(5
)3sin(3
)sin()( 000 +++ ttttvsq ωπ
ωπ
ωπ
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Filters
Filters process the sinusoid components of an inputsignal differently depending of the frequency ofsignal differently depending of the frequency of each component. Often, the goal of the filter is to retain the components in certain frequency rangesretain the components in certain frequency ranges and to reject components in other ranges.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Filters
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Transfer Functions
The transfer function H(f ) of the two-port filter is defined to be the ratio of the phasor output voltage to the phasor input voltage as a function of frequency:
( ) outV=fH ( )
inVf
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Transfer Functions
The magnitude of the transfer function showsThe magnitude of the transfer function shows how the amplitude of each frequency component is affected by the filter Similarly the phase of theis affected by the filter. Similarly, the phase of the transfer function shows how the phase of eachfrequency component is affected by the filterfrequency component is affected by the filter.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Transfer Functions
Magnitude Phase
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Example 6.1
402
10002
2000)402000cos(2)(
o
o
∠
==→+= Hzfttvin πππ
V
706)402)(303())(303(303)1000(
402
ooooo ∠=∠∠=∠=→=∠=
∠=
H inoutin
out
in
VVVV
V
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
)702000cos(6 o+= tvout π
Exercise 6.1
02
20002
4000)4000cos(2)(
o∠
==→= Hzfttvin πππ
V
604)02)(602())(602(602)2000(
02
ooooo ∠=∠∠=∠=→=∠=
∠=
H inoutin
out
in
VVVV
V
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
)604000cos(4 o+= tvout π
Exercise 6.1
201
30002
6000)206000cos(1)(
∠
==→−=
o
o Hzfttvin πππ
V
1100)201)(900())(900(900)3000(
201
∠=∠∠=∠=→=∠=
∠=
oooooH inoutin
out
in
VVVV
V
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
0)1106000cos(0 =+= otvout π
Graphic Equalizer
Filter with an adjustable transfer function
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Filter with an adjustable transfer function
Determining the output of a filterDetermining the output of a filter for an input with multiple
components:
1 D i h f d h1. Determine the frequency and phasor representation for each input component.
2. Determine the (complex) value of the transfer function for each component.p
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
3 Obtain the phasor for each output3. Obtain the phasor for each output component by multiplying the phasor for each input component by the correspondingeach input component by the corresponding transfer-function value.
4. Convert the phasors for the output components into time functions of various frequencies. Add these time functions to produce the output.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Example 6.2
o
o
033)(
)704000cos()2000cos(23)(
11 ∠==
−++=
inin
in
Vtv
tttv ππ
oo
o
701)704000cos()(
02)2000cos(2)(
)(
3
2
1
3
2
1
−∠=−=
∠==
inin
inin
inin
Vttv
Vttv
π
π
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
33 inin
Example 6.2
102)701)(602()2000(602)2000(
306)02)(303()1000(303)1000(
012)03)(04()0(4)0(
22
11
oooo
oooo
ooo
−∠=−∠∠==∠=
∠=∠∠==∠=
∠=∠∠===
VHVH
VHVH
VHVH
inout
inout
inout
)104000(2
)302000cos(6
12
0)70)(60()000(60)000(
2
1
33
o
o+=
=
tv
v
VV
out
out
inout
π
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
)104000cos(2)302000cos(612
)104000cos(2
321
3
oo
o
−+++=++=
−=
ttvvvv
tv
outoutoutout
out
ππ
π
Linear circuits behave as if they:
1. Separate the input signal into components p p g phaving various frequencies.
2. Alter the amplitude and phase of each component depending on its frequency.p p g q y
3. Add the altered components to produce p pthe output signal.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Determination of the Transfer Function
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
First-Order Low Pass Filter
R
in1
+=
VI
fCjR
in
π
11
2
⎟⎟⎞
⎜⎜⎛
⎟⎟⎞
⎜⎜⎛
+
VIV
fCjRfCjfCj
t
inout
πππ
1112122
⎟⎟⎟
⎠⎜⎜⎜
⎝+
⎟⎟⎠
⎜⎜⎝
==
V
IV
Half power
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
RCf
ffjfRCjfH B
Bin
out
ππ 21
)/(11
211)( =
+=
+==
VV Half power
frequency
First-Order Low Pass Filter
( ) ∠fH 011 o
( ) ( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛∠+
=+
=
BB
B
ffff
ffjfH
arctan11 2
⎠⎝
( )( )21
1
fffH = ( ) ⎟⎟
⎠
⎞⎜⎜⎝
⎛−=∠
fffH arctan
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
( )21 Bff+ ⎠⎝ Bf
First-Order Low Pass Filter
For low frequency signals the magnitude of the transfer function is unity and the phase is 0°. Low frequency signals are passed while high frequency signals are attenuated and
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
are passed while high frequency signals are attenuated and phase shifted.
Calculation of RC Low-Pass Output
tttvin )2000cos(5)200cos(5)20cos(5 ++= πππ
HzxRC
fB 100)1010(10002
12
16
=⎟⎞
⎜⎛
==−ππ
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
x )1010(2
2 ⎟⎠
⎜⎝ π
π
Calculation of RC Low-Pass Output
( )( )2
1
ffH = ( ) ⎟
⎠⎞
⎜⎝⎛−=∠100
arctan ffH
71.59950.0)100/10(1
1)10(05)20cos(511
oo −∠==∠==j
Htv inin π V
( )21001 f+ ⎠⎝100
2984099501)1000(05)2000cos(5
457071.0)100/100(1
1)100(05)200cos(5
)100/10(1)()(
22
11
oo
oo
∠∠
−∠=+
=∠==
+
Htv
jHtv
j
inin
inin
π
π
V
V
)71.520cos(975.471.5975.4)05)(71.59950.0()10(
29.840995.0)100/1000(1
)1000(05)2000cos(5
111
33
oooo −=−∠=∠−∠==
−∠=+
=∠==
tvH
jHtv
outinout
inin
π
π
VV
V
)29.842000cos(4975.029.844975.0)05)(29.840995.0()1000(
)45200cos(535.345535.3)05)(457071.0()100(
333
222
oooo
oooo
−=−∠=∠−∠==
−=−∠=∠−∠==
tvH
tvH
outinout
outinout
π
π
VV
VV
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
)29.842000cos(4975.0)45200cos(535.3)71.520cos(975.4)()()()(321
ooo −+−+−=++= ttttvtvtvtv outoutoutout πππ
Exercise 6.4
jfLRin
)2(+=
VI
RjfLR
R
jfLR
inout π
π
)2(
)2(
⎟⎟⎠
⎞⎜⎜⎝
⎛+
==
+
VIV
LRf
ffjLfjfLjRRfH
jfLR
BBin
out
πππ
π
2)/(11
21
12
)(
)2(
=+
=+
=+
==
⎠⎝ +
VV
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Rj1+
Exercise 6.5
111
)200/(11
)/(11)(
200102.01051
101025002
12
1)102cos(5)1000cos(5)40cos(10)( 236
4
+=
+=
=====++=−−
jfjfffH
HzxxxRC
fttttv
B
Bin
πππ
πππ
1
2.68857.1)2.683714.0)(05(2.683714.05.21
1)500(05)1000cos(5)(
711.595.9)711.5995.0)(010(711.5995.01.01
1)20(010)40cos(10)(
222
111
ooooo
ooooo
−∠=−∠∠=−∠=+
=∠==
−∠=−∠∠=−∠=+
=∠==
jHttv
jHttv
out
out
π
π
VV
VV
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.)9.88102cos(10.0)2.681000cos(85.1)71.540cos(95.9
85.881.0)85.881020)(05(85.8810205011)000,10(05)102cos(5)(
4
33
33
43
ooo
ooooo
−+−+−=
−∠=−∠∠=−∠=+
=∠== −−
tttv
xxj
Httv
out
out
πππ
π VV
Decibels, the Cascade Connection, and Logarithmic Frequency Scales
( ) ( )fHfH log20dB=
dB → decibelsdB → decibels
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Numbers greater than 1 are positivethan 1 are positive
Numbers smaller th 1 tithan 1 are negative
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
VVVVV)()()( 21
2
2
1
1
1
2
1
1
1
2 fHfHVV
VV
VV
VV
VV
fHin
out
in
out
out
out
in
out
in
out ====
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Cascaded Two-Port Networks
( ) ( ) ( )fHfHfH 21 ×=( ) ( ) ( )fff 21
( ) ( ) ( )fHfHfH +=( ) ( ) ( )dB2dB1dB
fHfHfH +=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
10=h
ff
One decade:
2=h
l
ff
One octave:
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
2=lf
One octave:
Logarithmic Frequency ScalesLogarithmic Frequency Scales
On a logarithmic scale, the variable is multiplied by a given factor for equalincrements of length along the axis.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
A decade is a range of frequencies for which the ratio of the highest frequency to the the ratio of the highest frequency to the lowest is 10.
⎟⎟⎠
⎞⎜⎜⎝
⎛=
1
2log decades ofNumber ff
⎠⎝ 1f
An octave is a two-to-one change in frequencyAn octave is a two-to-one change in frequency.
( )( ) ⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
2logloglog octaves ofNumber 12
1
22
ffff
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
( ) ⎠⎝⎠⎝ g1f
Exercise 6.6
Convert the magnitude of the transfer function H(f) into dB’s:
fH 50)( =
( ) dBfH dB 3450log20)( ==
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Exercise 6.7
623.510)()(log2015)()( 2015
==→== fHfHdBfHa dB )()(g)()(
30
fff dB
62.3110)()(log2030)()( 2030
==→== fHfHdBfHb dB
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Exercise 6.8
(a) What frequency is two octaves higher than 1000 Hz?(a) What frequency is two octaves higher than 1000 Hz?
1000log 2 ⎟
⎠⎞
⎜⎝⎛
⎞⎛f
f)2log(
1000g
1000log2 2
2
⎞⎛
⎠⎝=⎟⎠⎞
⎜⎝⎛=
f
f
)2log(21000
log
)2log(22
2 =⎟⎠⎞
⎜⎝⎛
f
f
000,4101000
101000
)2log(22
)2log(22
==
=
xf
f
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Exercise 6.8
(b) What frequency is three octaves lower than 1000 Hz?(b) What frequency is three octaves lower than 1000 Hz?
1000log1000 ⎟⎟
⎠
⎞⎜⎜⎝
⎛⎞⎛ f
1000
)2log(1000log3 1
12
⎞⎛
⎟⎠
⎜⎝=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
ff
1000
)2log(31000log
)2log(3
1=⎟⎟
⎠
⎞⎜⎜⎝
⎛f
1251000
101000
1
)2log(3
1
==
=
f
f
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
12510 )2log(31 ==f
Exercise 6.8
(c) What frequency is one decade lower than 1000 Hz?(c) What frequency is one decade lower than 1000 Hz?
⎞⎛ff
1010001000log1 =→⎟⎟⎠
⎞⎜⎜⎝
⎛=
Hzfff
100=⎠⎝
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Exercise 6.9
(a) What frequency is halfway between 100 and 1000 Hz(a) What frequency is halfway between 100 and 1000 Hz on a logarithmic frequency scale?
2)100l (3)1000log(
2)100log(==
average 5.22
23
2
=+
=
Hzff 2.316105.2)log( 5.2 ==→=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Exercise 6.9
(b) What frequency is halfway between 100 and 1000 Hz(b) What frequency is halfway between 100 and 1000 Hz on a linear frequency scale?
fHzf
Hzf1000100
2
1
==
HzHzHzf 5502
1001000=
+=
2
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.